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112 V NORMED SPACES

Suppose the theorem is proved for n - 1, and let H be the hyperplane
in E generated by a_l9..., ^_n_i; the inductive assumption implies that the
norm on H (induced by that of E) is equivalent to the norm sup |f,|;

Ki^n-l

hence H is complete (for both norms) and therefore closed in E (by (3.14.4)).
It follows from (5.8.1) that the mapping O^ + • •• 4- £_na_n) -» £„ is contin-
uous, and this, together with the inductive assumption, ends the proof
(by (3.20.4) and (5.4.2)).

(5.9.2) In a normed space E, let V be a closed subspace, W a finite dimensional
subspacei then V -f W is closed in E. In particular, any finite dimensional
subspace is closed in E.

We can use induction, on the dimension n of W, and therefore reduce
the proof to the case n = 1. Let W = Ra (resp. W = Gz); if aeV,
V + W = V and there is nothing to prove. If not, we can write any x e V + W
in the form x =f(x)a + y with y e V, and as V is a closed hyperplane in
V + W, /is continuous in V + W, by (5.8.1). Let (x_n) be a sequence of
points of V + W tending to a cluster point b of V + W (see (3.13.13));
write x_n -f(x_n)a 4- y_n. By (5.5.1), the sequence (f(x_n)) is a Cauchy sequence
in R (resp. C), hence tends to a limit A; therefore y_n = x_n ~-f(x_n)a tends to
b — A#; but as V is closed, the limit of (y_n) is in V, hence b eV + W. Q.E.D.
(See Section 6.5, Problem 2.)

(5.9.3) In a normed space E, let Y be a closed subspace of finite codimension
(i.e. having a finite dimensional algebraic supplement)', then any algebraic
supplement ofV is also a topological supplement.

Let W be an algebraic supplement of V in E; we use induction on the
dimension n of W, the result having been proved for n = 1 in (5.8.1). We can
write W = D + U where D is one-dimensional and U is (n — l)-dimensional
(direct sum); by (5.9.2), V + D is closed in E, hence U is a topological
supplement to V + D by the inductive assumption. In other words, E is
naturally homeomorphic to (V -f D) x U; by (5.8,1), V + D is naturally
homeomorphic to V x D, hence E is naturally homeomorphic to V x D x U.
Finally, as D x U is naturally homeomorphic to W, E is naturally homeo-
morphic to V x W. Q.E.D.

(5.9.4) (F. Riesz's theorem) A locally compact normed space E is finite
dimensional.



	112 V NORMED SPACES Suppose the theorem is proved for n - 1, and let H be the hyperplane in E generated by a_l9..., ^_n_i; the inductive assumption implies that the norm on H (induced by that of E) is equivalent to the norm sup \|f,\|; Ki^n-l hence H is complete (for both norms) and therefore closed in E (by (3.14.4)). It follows from (5.8.1) that the mapping O^ + • •• 4- £_na_n) -» £„ is contin- uous, and this, together with the inductive assumption, ends the proof (by (3.20.4) and (5.4.2)).

	(5.9.2) In a normed space E, let V be a closed subspace, W a finite dimensional subspacei then V -f W is closed in E. In particular, any finite dimensional subspace is closed in E. We can use induction, on the dimension n of W, and therefore reduce the proof to the case n = 1. Let W = Ra (resp. W = Gz); if aeV, V + W = V and there is nothing to prove. If not, we can write any x e V + W in the form x =f(x)a + y with y e V, and as V is a closed hyperplane in V + W, /is continuous in V + W, by (5.8.1). Let (x_n) be a sequence of points of V + W tending to a cluster point b of V + W (see (3.13.13)); write x_n -f(x_n)a 4- y_n. By (5.5.1), the sequence (f(x_n)) is a Cauchy sequence in R (resp. C), hence tends to a limit A; therefore y_n = x_n ~-f(x_n)a tends to b — A#; but as V is closed, the limit of (y_n) is in V, hence b eV + W. Q.E.D. (See Section 6.5, Problem 2.)

	(5.9.3) In a normed space E, let Y be a closed subspace of finite codimension (i.e. having a finite dimensional algebraic supplement)', then any algebraic supplement ofV is also a topological supplement. Let W be an algebraic supplement of V in E; we use induction on the dimension n of W, the result having been proved for n = 1 in (5.8.1). We can write W = D + U where D is one-dimensional and U is (n — l)-dimensional (direct sum); by (5.9.2), V + D is closed in E, hence U is a topological supplement to V + D by the inductive assumption. In other words, E is naturally homeomorphic to (V -f D) x U; by (5.8,1), V + D is naturally homeomorphic to V x D, hence E is naturally homeomorphic to V x D x U. Finally, as D x U is naturally homeomorphic to W, E is naturally homeo- morphic to V x W. Q.E.D.

	(5.9.4) (F. Riesz's theorem) A locally compact normed space E is finite dimensional.